www.underclassblog.com — A simple App to Backup and Restore SMS Messages. This is a Paid No-Ads version of the Ad-Supported Free App.

NOTE: The backup is created locally on the phone by default unless you configure it to upload to Email, Google Drive or Dropbox. You are responsible for backup files and the files are not sent to the developer in any case.

On Android 4.4+ devices, if you have performed the Restore and the messages don't appear in Hangouts then:

Open Hangouts and Disable SMS from Settings.

Exit the Hangouts app

Open Hangouts and Enable SMS from Settings.

APP FEATURES:

Backup SMS Messages in XML format.

Backup created locally on the device with options to automatically upload to Email, Google Drive or DropBox using the Add-On app.

Choose a scheduled time to automatically backup.

Backup format is independent of the Android version so the messages can be easily moved from one phone to another, irrespective of the Android version.

Option to select which conversations to backup.

View/Restore all Messages or only selected conversations.

Delete all SMS Messages on the Phone.

Email a backup file.

The XML can then be converted to other formats, and can also be viewed on a computer.

Tested on Android v1.5 to v4.4.2

On newer phones with inbuilt storage the default backup location will probably be the internal storage card and not the external.
This is because the phone reports the storage that way.

If you intend to do a factory reset on the phone, please make sure you save/email a copy of the backup outside the phone before doing it.
Lots more info at http://android.riteshsahu.com/apps/sms-backup-restore
FAQs can be found at http://bit.ly/d9t7Jk

Please start the app at least once after updates so that the scheduled backups start working.